Spectral Analysis Of The Quadratic Operator Pencils And The Infinite Dimensional Hamiltonian Operators

1990,Academician Wanxie Zhong conducted the elasticity to the infinite dimensional Hamiltonian system, established the new systematic methodology for the theory of elasticity.This method combined the elasticity with infinite di-mensional Hamiltonian operators,proposed the method of separation of variables based on Hamiltonian systems.It’s valuable to many questions.lt is wellknown of us,the method of separation of variables on account of Hamiltonian systems was based on the spectral theory of the infinite dimensional Hamiltonian operators and its completeness of the eigenfunction systems.In this dissertation,we centere on the spectral theory of the infinite dimensional Hamiltonian operators,discuss some questions as follows:Firstly,we conducted a class of quadratic operator pen-cils L(λ)=λ2M-iλK-Aand discuss the scatter of its spectrum.Secondly,we re-search the scatter of the spectrum of the non-negative infinite dimensional Hamil-tonian operators.Thirdly,the numerical range of the quadratic operator pencils is discussed. Fourthly,we discuss the spectrum of a class of the infinite dimensional Hamiltonian operators.The last,we discuss the essential spectrum of a class of infinite dimensional Hamiltonian operators.Firstly,we study the quadratic operator pencils L(λ)=λ2M-iλK-arising in researching the small vibrations of strings and beams.We give some sufficient conditions of the spectrum of this operator pencils.The sufficient conditions are when its spectrum located in the open upper half-plane and when its spectrum located in closed upper half-plane.And we give the condition of the spectrum which located in the open lower half-plane and the closed lower half-plane are only pure imaginary spectrum.Secondly,we study the scatter of the infinite dimensional Hamiltonian opera-tor.Wellknown of us, it is important a infinite dimensional Hamiltonian operator to be a generator of an operator semigroup.But if we try to solve this problem,we must solve the scatter of the spectrum of the operators which we discussed.,So we study the distributions of a class of non-negative infinite dimensional Hamil-tonian operator in chapter3.We give the sufficient conditions when its spectrum located in the open upper half-plane and when its spectrum located in the closed upper half-plane.Meanwhile we give the conditions of the infinite dimensional Hamiltonian operator whose spectrum is pure imaginary spectrum which located in the open lower half-plane and the closed lower half-plane.Thirdly,we discussed the numerical range of the quadratic operator pencils L(λ)=λ2M-iλK-A.Numerical range is a very useful method to estimate the spectrum of operator.To the numerical range of the quadratic operator pencils,we prove that its point spectrum is subset of its numerical range and its approximate spectrum is a subset of the closure of its numerical range.We prove its numerical is symmetric to the imaginary axis,but its not true to other operator pencils.We plot figures of numerical range of this quadratic operator pencils with Matlab software to discribe this result.Fourthly,by the special structure of the infinite dimensional Hamiltonian op-erator,we give a necessary and sufficient condition of its spectrum description.we know that the infinite dimensional Hamiltonian operator is an operator matrix,so we describe its spectrum with its entries operator.We obtain that its spectrum is equal to some combination of its entries.Lastly,we studied the essential spectrum of the infinite dimensional Hamilto-nian operator.We obtained the sufficient and necessary conditions of the infinite dimensional Hamiltonian operator to describe its spectrum.We give a description of the three classes of the essential spectrum of the infinite dimensional Hamil-tonian operator.All in all,We mainly study the spectrum of the quadratic operator pencil and the infinite dimensional Hamiltonian operator.This research provide some theoretical foundations for the application of the infinite dimensional Hamiltonian operator.